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Many Canadian banks (e.g. TD Canada Trust) split the same mutual fund into two options, one linked to the Canadian dollar and one linked to the US dollar. For example, the TD U.S. Index and the TD U.S. Index ($US).

What is the difference between these two mutual funds? Do I literally buy the $US one using U.S. dollars? If I expect the U.S. dollar to strengthen against the Canadian dollar (hypothetically), am I better off buying one over the other?

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The value of the fund and the change in that value is going to be related mostly to the contents of the fund. All the two versions are doing is basically allowing you to see a price both after and before exchange rates are taken into account.

If it's a US Index fund, investing in US stocks, then only a Canadian investor is likely to benefit or suffer from currency exchange rate changes. All a US investor would be doing is paying exchange rate fee's to their bank to change your money into CN$ Or the reverse if you live in Canada and buy the US version

For the Canadian investor this is effectively a 'international' fund of sorts, and currency risk is a part of that. As you will note that the fund has currently had a stronger return due to a weak Loonie, and that information is presented in BOTH versions of the fund, you could expect that for you, if the Loonie gets stronger, the performance (for you) of that fund to go down, no matter which version you invest in.

A Canadian investor who expects the Loonie to get stronger would be want to be investing in international instruments only to the extent they expected the return to outpace the changes in currency values, relative to what they could get returnwise in a domestic investment.

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A great writeup that goes into lots of detail is available at the Canadian Couch Potato blog. Excerpt:

The bottom line for Scott is that swapping his TD U.S. Index Fund for SPY won’t make a big difference if his account is in Canadian dollars. [...]

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